Subspace topology

Definition

We define the subspace topology as follows.

Given a topological space $$(X,\mathcal{J})$$ and any $$Y\subset X$$ we can define a topology on $$Y,\ (Y,\mathcal{J}_Y)$$ where $$\mathcal{J}_Y=\{Y\cap U|U\in\mathcal{J}\}$$

We may say " $$Y$$ is a subspace of $$X$$ (or indeed $$(X,\mathcal{J})$$ " to implicitly mean this topology.

Closed subspace

If $Y$ is a "closed subspace" of $(X,\mathcal{J})$ then it means that $Y$ is closed in $X$ and should be considered with the subspace topology.

Open subspace

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TODO: same as closed, but with the word "open"

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Open sets in open subspaces are open

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TODO: easy

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